Subsection CIS.EM Euler's Method Differential Equations with Initial Conditions

Subsection CIS.EM Differential Equations with Initial Conditions- Euler's Method

A differential equation (D.E.) is an equation involving derivatives in its statement, such as $\frac{dy}{dx} = x^2$ or $\frac{dy}{dx} = \frac {x^2}y$. The study of differential equations and their solutions is a study that often begins in the first semester of a calculus course and can continue through much of the study of mathematics and its application in almost every science.
For more on differential equations see The Sensible Calculus, IV. A. Differential Equations. Basic Concepts and Definitions.
In some situations a differential equation may be expressed using differentials!
For example, the equation $\frac{dy}{dx} = x^2$ might be expressed as $dy =x^2 dx$ and $\frac{dy}{dx} = \frac {x^2}y$ might be expressed as $dy =\frac{ x^2}y dx$ or as $y dy = x^2 dx$

A solution to a differential equation with one dependent variable is a differentiable (hence, continuous) function which satisfies the differential equation on an interval domain. The general solution to a differential equation is the family of all functions that are solutions to a given differential equation.

Often a differential equation has one or more additional conditions which a solution must satisfy as well, such as y(0) = 2 or y(3) = 4. Additional conditions often arise in applications from information given about the beginning of a situation ("initial conditions") or conditions that must be satisfied at the beginning and end of the process ("boundary conditions"). Such conditions restrict the selection of solutions to only a few particular members from the family of all solutions. A particular (or special) solution to a differential equation is a solution that satisfies any such additional conditions.
The interpretation of the derivative as a rate makes the study of differential equations a significant part of the study of rates in many contexts.
The visualization of rates and differential equations often uses a cartesian slope field. However mapping diagrams are not used in most calculus textbooks to visualize differential equations.
Before continuing with this section it would be helpful to review the materials in CCD.NA on the differential.

The following GeoGebra figure illustrates the connection between differential equations, mapping diagrams and cartesian visualizations of differential equations and their solutions.

Explanation: Use the slider to show examples with # var =1 [$\frac{dy}{dx} = x^2$] or # var = 2 [ $\frac{dy}{dx} = \frac {x^2}y$].
Change the example derivative, the value of a, or initial condition value for f(a) by making an entry in the appropriate box. The value of a can also be changed by using the slider. On the cartesian graph, moving the point labeled "Move me" will change both the value of a and the initial condition value for f(a).
Checking the boxes has the following effects:

Show Differential: Shows a second point, $x'=x+\Delta x$, on the mapping diagram determined by $\Delta x = dx$ on the slider. An arrow from $x'$ indicated the estimate of $f(x') \approx y + dy$ determined by the derivative, $\frac{dy}{dx} $. On the cartesian graph a line segment centered at the point labeled "Move me" of length $\Delta x$ visualizes a tangent line with slope $\frac{dy}{dx}$ at the point $(a, f(a)$. Moving the slider for $\Delta a$ will adjust the related fetures of the figure.
Show Solution: Shows a solution for the differential equation as determined by GeoGebra's computer algebra and the related cartesian graph as well as the value of the solution for $x'=x + \Delta x$ on the graph and the mapping diagram.
Show/Hide Slopefield: Shows the slope (tangent, direction) field for the differential equation. The slider N can be used to change the number of segments shown in the field and their lengths.

Euler's method is a numerical method to estimate the solution of an initial value problem for a differential equation. It is often visualized with direction (tangent) fields on graphs in a beginning calculus course. An alternative that presents a different connection to the repeated use of the differential is visualized with mapping diagrams.

CIS.EM.TMD. Euler's Method with Tables and Mapping Diagrams.